Related papers: Error bounds for PDE-regularized learning
We consider a linear elliptic partial differential equation (PDE) with a generic uniformly bounded parametric coefficient. The solution to this PDE problem is approximated in the framework of stochastic Galerkin finite element methods. We…
Models incorporating uncertain inputs, such as random forces or material parameters, have been of increasing interest in PDE-constrained optimization. In this paper, we focus on the efficient numerical minimization of a convex and smooth…
We propose a neural network-based meta-learning method to efficiently solve partial differential equation (PDE) problems. The proposed method is designed to meta-learn how to solve a wide variety of PDE problems, and uses the knowledge for…
Many engineering and scientific fields have recently become interested in modeling terms in partial differential equations (PDEs) with neural networks, which requires solving the inverse problem of learning neural network terms from…
Developing efficient numerical algorithms for the solution of high dimensional random Partial Differential Equations (PDEs) has been a challenging task due to the well-known curse of dimensionality. We present a new solution framework for…
This article is an overview of supervised machine learning problems for regression and classification. Topics include: kernel methods, training by stochastic gradient descent, deep learning architecture, losses for classification,…
We study problem-dependent rates, i.e., generalization errors that scale near-optimally with the variance, the effective loss, or the gradient norms evaluated at the "best hypothesis." We introduce a principled framework dubbed "uniform…
We use the ideas of goal-oriented error estimation and adaptivity to design and implement an efficient adaptive algorithm for approximating linear quantities of interest derived from solutions to elliptic partial differential equations…
In [1] we consider an optimal control problem subject to a semilinear elliptic PDE together with its variational discretization, where we provide a condition which allows to decide whether a solution of the necessary first order conditions…
Numerically solving high-dimensional partial differential equations (PDEs) is a major challenge. Conventional methods, such as finite difference methods, are unable to solve high-dimensional PDEs due to the curse-of-dimensionality. A…
To capture and simulate geometric surface evolutions, one effective approach is based on the phase field methods. Among them, it is important to design and analyze numerical approximations whose error bound depends on the inverse of the…
Computational approaches to PDE-constrained optimization under uncertainty may involve finite-dimensional approximations of control and state spaces, sample average approximations of measures of risk and reliability, smooth approximations…
This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse…
Stochastic differential equations (SDEs) have been shown recently to characterize well the dynamics of training machine learning models with SGD. When the generalization error of the SDE approximation closely aligns with that of SGD in…
Topological optimization finds a material density distribution minimizing a functional of the solution of a partial differential equation (PDE), subject to a set of constraints (typically, a bound on the volume or mass of the material).…
In this paper, we study the statistical limits of deep learning techniques for solving elliptic partial differential equations (PDEs) from random samples using the Deep Ritz Method (DRM) and Physics-Informed Neural Networks (PINNs). To…
In this paper, we study a learning problem in which a forecaster only observes partial information. By properly rescaling the problem, we heuristically derive a limiting PDE on Wasserstein space which characterizes the asymptotic behavior…
We consider Galerkin finite element methods for semilinear stochastic partial differential equations (SPDEs) with multiplicative noise and Lipschitz continuous nonlinearities. We analyze the strong error of convergence for spatially…
In this paper we consider Deep Neural Networks (DNNs) with a smooth activation function as surrogates for high-dimensional functions that are somewhat smooth but costly to evaluate. We consider the standard (non-periodic) DNNs as well as…
Partial differential equations (PDEs) are widely used across the physical and computational sciences. Decades of research and engineering went into designing fast iterative solution methods. Existing solvers are general purpose, but may be…